Joint work with Richard Baltensperger.
The global pseudospectral method of lines for solving hyperbolic or parabolic differential equations for a function u(x,t) of the spatial variable x and the time variable t can be described as follows: One chooses interpolation points (nodes) xi in the spatial interval, replaces u for every t by a polynomial û interpolating between the xi's, inserts û into the equation and collocates at the xi's. This leads to a system of ODEs in t that is then solved by a time marching technique. In the latter, the vectors of values of û are multiplied by differentiation matrices D whose entries are the derivatives of the fundamental Lagrange polynomials at the nodes.
For precise and well-conditioned interpolation, the xi's must be chosen in such a way that their preimages on the circle by the application arccos are almost equidistant (e.g., Chebyshev or Legendre points), which make them concentrate near the extremities of the interval. This results in ill-conditioned differentiation and highly nonnormal D's which for stability require smaller time steps than difference or element methods using equidistant nodes.
For the Chebyshev case, Kosloff and Tal-Ezer in 1993 suggested to shift the points toward a more regular distribution by means of a conformal map. In view of the chain rule, the corresponding change of variable results in differentiation matrices that are sums of products of the D's with diagonal matrices of values of derivatives of the map. These new matrices are closer to normality than the D's and therefore increase the timestep necessary for the stability of the time marching technique.
In our talk we will show how the same can be achieved by replacing the interpolating polynomial by some linear rational interpolant, i.e., a rational interpolant whose denominator is independent of the interpolated function.